   # Solving a Differential Equation Solve the differential equation given that y 1 and y 2 are solutions of the corresponding homogeneous equation. (a) x 2 y ″ − x y ′ + y = 4 x ln x y 1 = x and y 2 = x ln x (b) x 2 y ″ + x y ′ + 4 y = sin ( ln x ) y 1 = sin ( ln x 2 ) and y 2 = cos ( ln x 2 ) ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

#### Solutions

Chapter
Section ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16.3, Problem 38E
Textbook Problem
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## Solving a Differential Equation Solve the differential equation given that y1 and y2 are solutions of the corresponding homogeneous equation.(a) x 2 y ″ − x y ′ + y = 4 x ln x y 1 = x   and   y 2 = x ln x (b) x 2 y ″ + x y ′ + 4 y = sin ( ln x ) y 1 = sin ( ln x 2 ) and   y 2 = cos ( ln x 2 )

(a)

To determine

The solution of the equation x2yxy+y=4xlnx given that y1=xandy2=xlnx are the solutions of the corresponding nonhomogeneous equation.

### Explanation of Solution

Given information:

The differential equation x2yxy+y=4xlnx.

The solution of the corresponding nonhomogeneous equation y1=xandy2=xlnx.

As the solution of the nonhomogeneous equationis y1=xandy2=xlnx, the particular solution is:

yp=Ay1+By2=Ax+Bxlnx

A and B are constants coefficients.

Let y=xm.

Differentiate the above equation with respect to y.

y=mxm1

Again differentiate the above equation with respect to y.

y=m(m1)xm2

Substitute the values of y,yandy in the homogeneous equation x2yxy+y=0 and simplify:

x2m(m1)xm2xmx

(b)

To determine

The solution of the equation x2y+xy+4y=sin(lnx) given that y1=sin(lnx2)andy2=cos(lnx2) are the solutions of the corresponding nonhomogeneous equation.

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